A sonometer wire is stretched by hanging a metal bob. The fundamental frequency of vibration of wire is ' \(n_1\) '. When the bob is completely immersed in water, the frequency of vibration of wire becomes ' \(n_2\) '. The relative density of the metal of the bob is

\(\mathrm{n}=\frac{1}{2 \mathrm{~L}} \sqrt{\frac{\mathrm{~T}}{\mu}}\)
The sonometer wire is stretched by a metal bob,
\(\mathrm{T}=\mathrm{mg}=\mathrm{W}\)
where W is weight of metal bob
\(\begin{array}{ll}
\therefore \quad & \text { From(i), } \\
& \mathrm{n}_1=\frac{1}{2 \mathrm{~L}} \sqrt{\frac{\mathrm{~W}_1}{\mu}}, \quad \mathrm{n}_2=\frac{1}{2 \mathrm{~L}} \sqrt{\frac{\mathrm{~W}_2}{\mu}} \\
\therefore \quad & \frac{\mathrm{~W}_1}{\mathrm{~W}_2}=\frac{\mathrm{n}_1^2}{\mathrm{n}_2^2} \\
& \text { Relative density }(\sigma)=\frac{\text { weight in air }}{\text { loss of weight in water }} \\
\therefore \quad & \frac{\mathrm{W}_1}{\mathrm{~W}_1-\mathrm{W}_2}=\frac{\mathrm{n}_1^2}{\mathrm{n}_1^2-\mathrm{n}_2^2}
\end{array}\)